Why can't a malicious site obtain a CSRF token via GET before attacking?

If I understand correctly, in a CSRF attack a malicious website A tells my browser to send a request to site B. My browser will automatically include my B cookies in that request. Although A cannot see those cookies, if I'm already authenticated in B the request will look legit, and whatever action was asked will be successfully performed. To avoid this, every time that I visit a page of B containing a form, I receive a CSRF token. This token is associated to my session, so if I make a POST to B I MUST include such token; otherwise B rejects my request. The benefit of this scheme is that A will not have access to that token.

I have two questions:

Is the above description correct?

If so, why can't site A first tell my browser to send a GET to B, obtain the CSRF token from the response, and then use the token to send now a POST to B? Notice that the token will be valid and associated to my session, as the GET also contains all my B cookies.

If site A tells your browser to go to B and get the token, that's fine, but as it is a cross-domain request, A will not have access to the token in Javascript (this is a browser feature). So when A tells your browser to go back to B and actually do something, it still cannot include the token in the request.

That is, unless B set the token as a cookie. Evidently, that would be flawed, because the token cookie would also be sent, thus negating any protection. So the token in this case must be sent as either a form value or a request header (or something else that is not sent automatically like a cookie).

This also means that if B is vulnerable to cross-site scripting, it is also vulnerable to CSRF, because the token can then be stolen, but CSRF is the smaller problem then. :)